Expansivity and unique shadowing

Chris Good, Sergio Macias, Jonathan Meddaugh, Joel Mitchell, Joe Thomas

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Abstract

Let f:X→X be a continuous function on a compact metric space. We show that shadowing is equivalent to backwards shadowing and two-sided shadowing when the map f is onto. Using this we go on to show that, for expansive surjective maps the properties shadowing, two-sided shadowing, s-limit shadowing and two-sided s-limit shadowing are equivalent. We show that is positively expansive and has shadowing if and only if it has unique shadowing (i.e.\ each pseudo-orbit is shadowed by a unique point), extending a result implicit in Walter's proof that positively expansive maps with shadowing are topologically stable. We use the aforementioned result on two-sided shadowing to find an equivalent characterisation of shadowing and expansivity and extend these results to the notion of n-expansivity due to Morales.
Original languageEnglish
Pages (from-to)1-15
JournalProceedings of the American Mathematical Society
Volume2020
DOIs
Publication statusPublished - 25 Nov 2020

Keywords

  • Expansive
  • shadowing
  • pseudo-orbit
  • s-limit shadowing

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